4,540 research outputs found
Fractal Weyl laws and wave decay for general trapping
We prove a Weyl upper bound on the number of scattering resonances in strips
for manifolds with Euclidean infinite ends. In contrast with previous results,
we do not make any strong structural assumptions on the geodesic flow on the
trapped set (such as hyperbolicity) and instead use propagation statements up
to the Ehrenfest time. By a similar method we prove a decay statement with high
probability for linear waves with random initial data. The latter statement is
related heuristically to the Weyl upper bound. For geodesic flows with positive
escape rate, we obtain a power improvement over the trivial Weyl bound and
exponential decay up to twice the Ehrenfest time.Comment: 36 pages, 5 figures; minor revision
Skinning maps
Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We
prove that there are upper and lower bounds on the diameter of the skinning map
of M that depend only on the volume of the hyperbolic structure with totally
geodesic boundary, answering a question of Y. Minsky. This is proven via a
filling theorem, which states that as one performs higher and higher Dehn
fillings, the skinning maps converge uniformly on all of Teichmuller space.
We also exhibit manifolds with totally geodesic boundaries whose skinning
maps have diameter tending to infinity, as well as manifolds whose skinning
maps have diameter tending to zero (the latter are due to K. Bromberg and the
author).
In the final section, we give a proof of Thurston's Bounded Image Theorem.Comment: 50 pages, 4 figures. v3. Major revision incorporating referees'
comments. To appear in the Duke Mathematical Journal. v2. Cosmetic changes,
minor corrections, inclusion of theorem with K. Bromber
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